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ABSTRACT
For recovering an unknown heat source F(x, t) = G(x) + H(t) in the heat conduction equation, we develop a homogenized function method and the expansion methods by polynomials or eigenfunctions, which can solve the inverse heat source recovery problem by using collocation technique. Because the initial condition/boundary conditions/supplementary condition are satisfied automatically and a rectangular differencing technique is developed, a middle-scale linear system is sufficient to determine the expansion coefficients. After deriving a multiscale postconditioning matrix, the present methods converge very quickly, and are accurate and stable against large noise, as verified by numerical tests.
Nomenclature
| A | = | coefficient matrix in Eq. (26) |
| b | = | right-hand side in Eq. (26) |
| b1 | = | := AT b |
| B | = | new coefficient matrix in Eq. (36) |
| cij | = | coefficients in Pascal polynomial and eigenfunctions |
| ck | = | vectorized of cij |
| c | = | n-dimensional vector of coefficients |
| D | = | := AT A |
| f(x) | = | initial temperature |
| F(x, t) | = | heat source |
| g(x) | = | final temperature |
| G(x) | = | space-dependent heat source |
| H(t) | = | time-dependent heat source |
| ℓ | = | length of rod |
| m | = | m − 1 the highest order of Pascal polynomial |
| m | = | m × m terms of eigenfunctions |
| m1 | = | number of collocation points on a horizontal line |
| m2 | = | number of collocation points on a vertical line |
| n | = | := m(m + 1)/2 or m2 |
| P | = | postconditioning matrix in Eq. (35) |
| R(x) | = | random function |
| R(i) | = | random number |
| t | = | time |
| tf | = | final time |
| ti | = | := itf/m2 |
| u(x, t) | = | temperature |
| uℓ(t) | = | right-boundary temperature |
| u0(t) | = | left-boundary temperature |
| v(x, t) | = | := u(x, t) − w(x, t) |
| w(x, t) | = | homogenized function |
| x | = | space variable |
| xi | = | := iℓ/m1 |
| ϵ | = | converegnce criterion |
| σ | = | level of noise |
| Subscripts and superscripts | = | |
| i | = | index |
| j | = | index |
| k | = | index |
| m | = | index |
| T | = | transpose |
Nomenclature
| A | = | coefficient matrix in Eq. (26) |
| b | = | right-hand side in Eq. (26) |
| b1 | = | := AT b |
| B | = | new coefficient matrix in Eq. (36) |
| cij | = | coefficients in Pascal polynomial and eigenfunctions |
| ck | = | vectorized of cij |
| c | = | n-dimensional vector of coefficients |
| D | = | := AT A |
| f(x) | = | initial temperature |
| F(x, t) | = | heat source |
| g(x) | = | final temperature |
| G(x) | = | space-dependent heat source |
| H(t) | = | time-dependent heat source |
| ℓ | = | length of rod |
| m | = | m − 1 the highest order of Pascal polynomial |
| m | = | m × m terms of eigenfunctions |
| m1 | = | number of collocation points on a horizontal line |
| m2 | = | number of collocation points on a vertical line |
| n | = | := m(m + 1)/2 or m2 |
| P | = | postconditioning matrix in Eq. (35) |
| R(x) | = | random function |
| R(i) | = | random number |
| t | = | time |
| tf | = | final time |
| ti | = | := itf/m2 |
| u(x, t) | = | temperature |
| uℓ(t) | = | right-boundary temperature |
| u0(t) | = | left-boundary temperature |
| v(x, t) | = | := u(x, t) − w(x, t) |
| w(x, t) | = | homogenized function |
| x | = | space variable |
| xi | = | := iℓ/m1 |
| ϵ | = | converegnce criterion |
| σ | = | level of noise |
| Subscripts and superscripts | = | |
| i | = | index |
| j | = | index |
| k | = | index |
| m | = | index |
| T | = | transpose |